Applications of Derivatives — Part 1: Rates, monotonicity, tangents and normals: visual lesson, ISC Class 12 Mathematics

Lesson

Applications of Derivatives, Part 1

A ripple spreads, a tank fills, a road bends: derivatives describe how fast and in which direction things change.

In this lesson you will

Use the derivative as a rate of change and solve related-rate problems
Find where a function is strictly increasing or strictly decreasing
Write the equations of the tangent and the normal to a curve at a point
Find the angle between two intersecting curves

Part 1 at a glance.

Concept

The derivative as a rate of change

If $y = f(x)$ , then $\dfrac{dy}{dx}$ is the rate of change of $y$ with respect to $x$ . When both depend on time, $\dfrac{dy}{dt} = \dfrac{dy}{dx}\cdot\dfrac{dx}{dt}$ .

A derivative measures how fast one quantity changes with another.
Quantity	Rate	Units
Area $A$ of a ripple	$\dfrac{dA}{dt}$	cm $^2$ per second
Edge $x$ of a cube	$\dfrac{dx}{dt}$	cm per second
Cost $C$ of $x$ items	$\dfrac{dC}{dx}$ (marginal cost)	rupees per item

Worked example

Worked example 1: a spreading ripple

The radius of a circular ripple grows at $4$ cm/s. How fast is its area growing when $r = 8$ cm?

$A = \pi r^2$
$\dfrac{dA}{dt} = 2\pi r\dfrac{dr}{dt}$
$= 2\pi(8)(4) = 64\pi$ cm $^2$ /s

The same ripple grows faster in area as it gets bigger.
Radius $r$ (cm)	$\dfrac{dA}{dt} = 2\pi r \cdot 4$
$4$	$32\pi$
$8$	$64\pi$
$12$	$96\pi$

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